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Polygons are a ubiquitous modeling primitive in computer graphics. However, modeling with polygons is problematic for highly faceted approximations to smooth surfaces. The sheer size of these approximations makes them impossible to manipulate directly. Subdivision solves this problem by representing a smooth shape in terms of a coarse polygonal model. The subdivision rules used during this refinement process depend only on the initial model's topological connectivity and yield surfaces with guaranteed smoothness. Subdivision schemes are either interpolating or approximating. The averaging methods we've described are approximating in that the surfaces don't interpolate the original surface's vertices. Interpolating methods interpolate the vertices of the original surface, giving the user a more intuitive feel of the final surface shape.